Best Known (41, 41+7, s)-Nets in Base 3
(41, 41+7, 59053)-Net over F3 — Constructive and digital
Digital (41, 48, 59053)-net over F3, using
- 32 times duplication [i] based on digital (39, 46, 59053)-net over F3, using
- net defined by OOA [i] based on linear OOA(346, 59053, F3, 7, 7) (dual of [(59053, 7), 413325, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(346, 177160, F3, 7) (dual of [177160, 177114, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(346, 177160, F3, 7) (dual of [177160, 177114, 8]-code), using
- net defined by OOA [i] based on linear OOA(346, 59053, F3, 7, 7) (dual of [(59053, 7), 413325, 8]-NRT-code), using
(41, 41+7, 88581)-Net over F3 — Digital
Digital (41, 48, 88581)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(348, 88581, F3, 2, 7) (dual of [(88581, 2), 177114, 8]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(346, 88580, F3, 2, 7) (dual of [(88580, 2), 177114, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(346, 177160, F3, 7) (dual of [177160, 177114, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(346, 177160, F3, 7) (dual of [177160, 177114, 8]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(346, 88580, F3, 2, 7) (dual of [(88580, 2), 177114, 8]-NRT-code), using
(41, 41+7, large)-Net in Base 3 — Upper bound on s
There is no (41, 48, large)-net in base 3, because
- 5 times m-reduction [i] would yield (41, 43, large)-net in base 3, but